1,4

Each triangular layer of the unique tetrahedron begins with 1, never uses any value other than 1 which has occurred already on this or earlier levels, always uses the least available integer such that the sum of each two consecutive entries is a prime. The number of values of the n-th level is the n-th triangular number A000217(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n. The number of values through the n-th level is the n-th tetrahedral number A000292(n) = C(n+2,3) = n(n+1)(n+2)/6.

R. K. Guy, Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 106, 1994.

Kenney, M. J. "Student Math Notes." NCTM News Bulletin. Nov. 1986.

Eric Weisstein's World of Mathematics, Prime Triangle.

a(n) flattens the 3-D table so that level 1 (the apex, with only the value 1) occures first, then level 2 (with values 1, 1, 2), then level 3 ... and for each level, reads that triangle by rows.

Tetrahedron begins

=================

1

=================

1

1..2

=================

1

1..4

1..6..5

=================

1

1.10

1.12..7

1.16..3..8

=================

1

1.18

1.22..9

1.28.13.24

1.30.11.20.17

=================

srch := proc(a) local res ; res := 2 ; while true do if isprime(res+op(-1, a)) and not ( res in a ) then RETURN(res) ; fi ; res := res+1 ; od ; end: a := [] ; for lvl from 1 to 10 do for row from 1 to lvl do for col from 1 to row do if col = 1 then anxt := 1 ; else anxt := srch(a) ; fi ; printf("%d, ", anxt) ; a := [op(a), anxt] ; od ; od ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 13 2007

Cf. A000040, A000217, A000292, A036440 Number of ways of arranging row n of the Prime Pyramid, A051239, A051237 Lexicographically earliest Prime Pyramid, read by rows.

Sequence in context: A112987 A125138 A021477 this_sequence A099020 A089688 A092479

Adjacent sequences: A124936 A124937 A124938 this_sequence A124940 A124941 A124942

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 13 2006

Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 13 2007